Footprint Model of Hsieh et al. (2000)
The footprint model of Hsieh et al. (2000) is an approximate analytical model developed to estimate scalar flux footprint in thermally stratified atmospheric surface layer (ASL) flows.
The proposed model was based on a combination of Lagrangian stochastic dispersion model results and dimensional analysis.
The main advantage of this model is its ability to analytically relate atmospheric stability, measurement height, and surface roughness length to flux and footprint.
Flux estimation by the proposed model was shown to be in good agreement with those calculated by detailed Eulerian and Lagrangian models. The model formulation and comparison with field data are described in Hsieh, C.I., G.G. Katul, and T.W. Chi, 2000, "An approximate analytical model for footprint estimation of scalar fluxes in thermally stratified atmospheric flows", Advances in Water Resources, 23, 765-772.Full Manuscript in PDF.
The model can be used as a MATLAB function: [Fc, Fp, L, xp, x]=footprint_hsieh(ustar, H, Ta, zm, zo)
Model Output Are:
Fc(x) = Cumulative source contribution with distance (fraction) [Note: as x--> +infinity, Fc(x)--> 1]
fp (x)= Source-weight function [Note: fp(x)=dFc(x)/dx]
L = Obukhov length (m)
xp = Peak distance from measuring point to the maximum contributing source area (m) [Note: xp is defined as the x at which fp(x) is maximum]
x = Distance vector from the tower along the mean wind direction used in the computation of Fc, fp (m)
F2H=Fetch to Height ratio (for 90% of flux recovery, i.e. 100:1 or 20:1)[Note: Footprint is often calculated as F2H x zm, zm is the instrument height]
Model Input Are:
ustar = friction velocity (m s-1)
H = Sensible heat (W m-2)
Ta = Mean air temperature (oC)
zm = Instrument height (m)
zo = Momentum roughness height (m)
EXAMPLE: A demo MATLAB program with a sample data set from the Duke Forest grass-covered forest clearing is also included.
The example also illustrates polar graphics in MATLAB and the large differences in footprint values between unstable and stable ASL flows (see sample figure). Note, the unstable fetch is a factor of 10 smaller than its stable counterpart for this data set. The data set, which includes more than 11,0000 30 minute runs, was collected using a CSAT3 (Campbell Scientific Inc, Logan, Utah) triaxial sonic anemometer situated at 3.8 m (=zm) above a 0.5 m(=h) tall grass surface. The zero-plane displacement (d) and momentum roughness height (z 0) were assumed as 0.67 h, and 0.1 h, respectively. The sample file is a tab-delimited ascii data set and includes 7 columns: DOY, Time, ustar, Ta, H, Rn,WD, where DOY is the day of year, Time is in hours and minutes (HHMM), ustar is the friction velocity (m s-1), Ta is the mean air temperature (oC), H is the sensible heat flux (W m-2), Rn is the net radiation (W m-2), and WD is wind direction.